# -*- coding: utf-8 -*-
"""
Created on Thu Oct  5 16:44:23 2017

@author: Christian Bender

This python library contains some useful functions to deal with
prime numbers and whole numbers.

Overview:

isPrime(number)
sieveEr(N)
getPrimeNumbers(N)
primeFactorization(number)
greatestPrimeFactor(number)
smallestPrimeFactor(number)
getPrime(n)
getPrimesBetween(pNumber1, pNumber2) 

----

isEven(number)
isOdd(number)
gcd(number1, number2)  // greatest common divisor
kgV(number1, number2)  // least common multiple
getDivisors(number)    // all divisors of 'number' inclusive 1, number
isPerfectNumber(number)

NEW-FUNCTIONS

simplifyFraction(numerator, denominator)
factorial (n) // n!
fib (n) // calculate the n-th fibonacci term.

-----

goldbach(number)  // Goldbach's assumption

"""

def isPrime(number):
    """
        input: positive integer 'number'
        returns true if 'number' is prime otherwise false.
    """
    import math # for function sqrt
    
    # precondition
    assert isinstance(number,int) and (number >= 0) , \
    "'number' must been an int and positive"
    
    status = True
    
    # 0 and 1 are none primes. 
    if number <= 1:
        status = False
    
    for divisor in range(2,int(round(math.sqrt(number)))+1):
        
        # if 'number' divisible by 'divisor' then sets 'status'
        # of false and break up the loop. 
        if number % divisor == 0:
            status = False
            break
    
    # precondition
    assert isinstance(status,bool), "'status' must been from type bool"    
    
    return status

# ------------------------------------------

def sieveEr(N):
    """
        input: positive integer 'N' > 2
        returns a list of prime numbers from 2 up to N.
        
        This function implements the algorithm called
        sieve of erathostenes.         
        
    """
    
    # precondition
    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
    
    # beginList: conatins all natural numbers from 2 upt to N
    beginList = [x for x in range(2,N+1)]

    ans = [] # this list will be returns.     
    
    # actual sieve of erathostenes
    for i in range(len(beginList)):
        
        for j in range(i+1,len(beginList)):
            
            if (beginList[i] != 0) and \
            (beginList[j] % beginList[i] == 0):
                beginList[j] = 0
    
    # filters actual prime numbers.           
    ans = [x for x in beginList if x != 0]
    
    # precondition
    assert isinstance(ans,list), "'ans' must been from type list"    
    
    return ans
    

# --------------------------------

def getPrimeNumbers(N):
    """
        input: positive integer 'N' > 2
        returns a list of prime numbers from 2 up to N (inclusive)
        This function is more efficient as function 'sieveEr(...)'
    """    
    
    # precondition
    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
    
    ans = []    
    
    # iterates over all numbers between 2 up to N+1 
    # if a number is prime then appends to list 'ans'
    for number in range(2,N+1):
        
        if isPrime(number):
            
            ans.append(number)
   
    # precondition
    assert isinstance(ans,list), "'ans' must been from type list"
        
    return ans


# -----------------------------------------
    
def primeFactorization(number):
    """
        input: positive integer 'number' 
        returns a list of the prime number factors of 'number'
    """

    import math    # for function sqrt
    
    # precondition
    assert isinstance(number,int) and number >= 0, \
    "'number' must been an int and >= 0"
    
    ans = [] # this list will be returns of the function.

    # potential prime number factors.

    factor = 2    

    quotient = number
    
    
    if number == 0 or number == 1:
        
        ans.append(number)
        
    # if 'number' not prime then builds the prime factorization of 'number'    
    elif not isPrime(number):
    
        while (quotient != 1):
            
            if isPrime(factor) and (quotient % factor == 0):
                    ans.append(factor)
                    quotient /= factor
            else:
                    factor += 1
    
    else:
        ans.append(number)
    
    # precondition
    assert isinstance(ans,list), "'ans' must been from type list"    
    
    return ans
    

# -----------------------------------------
    
def greatestPrimeFactor(number):
    """
        input: positive integer 'number' >= 0
        returns the greatest prime number factor of 'number'
    """
    
    # precondition
    assert isinstance(number,int) and (number >= 0), \
    "'number' bust been an int and >= 0"
    
    ans = 0    
    
    # prime factorization of 'number'
    primeFactors = primeFactorization(number)

    ans = max(primeFactors)     
    
    # precondition
    assert isinstance(ans,int), "'ans' must been from type int"    
    
    return ans
    

# ----------------------------------------------
    
    
def smallestPrimeFactor(number):
    """
        input: integer 'number' >= 0
        returns the smallest prime number factor of 'number'
    """
    
    # precondition
    assert isinstance(number,int) and (number >= 0), \
    "'number' bust been an int and >= 0"
    
    ans = 0    
    
    # prime factorization of 'number'
    primeFactors = primeFactorization(number)
    
    ans = min(primeFactors)

    # precondition
    assert isinstance(ans,int), "'ans' must been from type int"    
    
    return ans
    
    
# ----------------------
    
def isEven(number):
    """
        input: integer 'number'
        returns true if 'number' is even, otherwise false.
    """   

    # precondition
    assert isinstance(number, int), "'number' must been an int" 
    assert isinstance(number % 2 == 0, bool), "compare bust been from type bool"
    
    return number % 2 == 0
    
# ------------------------
    
def isOdd(number):
    """
        input: integer 'number'
        returns true if 'number' is odd, otherwise false.
    """   

    # precondition
    assert isinstance(number, int), "'number' must been an int" 
    assert isinstance(number % 2 != 0, bool), "compare bust been from type bool"
    
    return number % 2 != 0
    
# ------------------------
    
    
def goldbach(number):
    """
        Goldbach's assumption
        input: a even positive integer 'number' > 2
        returns a list of two prime numbers whose sum is equal to 'number'
    """
    
    # precondition
    assert isinstance(number,int) and (number > 2) and isEven(number), \
    "'number' must been an int, even and > 2"
    
    ans = [] # this list will returned
    
    # creates a list of prime numbers between 2 up to 'number'
    primeNumbers = getPrimeNumbers(number)
    lenPN = len(primeNumbers)    

    # run variable for while-loops.
    i = 0
    j = 1
    
    # exit variable. for break up the loops
    loop = True
    
    while (i < lenPN and loop):
        
        j = i+1
        
        
        while (j < lenPN and loop):
            
            if primeNumbers[i] + primeNumbers[j] == number:
                loop = False
                ans.append(primeNumbers[i])
                ans.append(primeNumbers[j])
                
            j += 1

        i += 1
        
    # precondition
    assert isinstance(ans,list) and (len(ans) == 2) and \
    (ans[0] + ans[1] == number) and isPrime(ans[0]) and isPrime(ans[1]), \
    "'ans' must contains two primes. And sum of elements must been eq 'number'"
        
    return ans
    
# ----------------------------------------------

def gcd(number1,number2):
    """
        Greatest common divisor
        input: two positive integer 'number1' and 'number2'
        returns the greatest common divisor of 'number1' and 'number2'
    """
    
    # precondition
    assert isinstance(number1,int) and isinstance(number2,int) \
    and (number1 >= 0) and (number2 >= 0), \
    "'number1' and 'number2' must been positive integer."

    rest = 0    
    
    while number2 != 0:
        
        rest = number1 % number2
        number1 = number2
        number2 = rest

    # precondition
    assert isinstance(number1,int) and (number1 >= 0), \
    "'number' must been from type int and positive"    
    
    return number1
    
# ----------------------------------------------------
    
def kgV(number1, number2):
    """
        Least common multiple
        input: two positive integer 'number1' and 'number2'
        returns the least common multiple of 'number1' and 'number2'
    """
    
    # precondition
    assert isinstance(number1,int) and isinstance(number2,int) \
    and (number1 >= 1) and (number2 >= 1), \
    "'number1' and 'number2' must been positive integer."
    
    ans = 1 # actual answer that will be return.
     
    # for kgV (x,1)
    if number1 > 1 and number2 > 1:
        
        # builds the prime factorization of 'number1' and 'number2'
        primeFac1 = primeFactorization(number1)
        primeFac2 = primeFactorization(number2)
        
    elif number1 == 1 or number2 == 1:
        
        primeFac1 = []
        primeFac2 = []
        ans = max(number1,number2)
    
    count1 = 0
    count2 = 0
     
    done = [] # captured numbers int both 'primeFac1' and 'primeFac2'
    
    # iterates through primeFac1
    for n in primeFac1:
        
        if n not in done:
        
            if n in primeFac2:
            
                count1 = primeFac1.count(n)
                count2 = primeFac2.count(n)
            
                for i in range(max(count1,count2)):
                    ans *= n
        
            else:
                
                count1 = primeFac1.count(n)
                
                for i in range(count1):
                    ans *= n
                    
            done.append(n)
    
    # iterates through primeFac2
    for n in primeFac2:
        
        if n not in done:
            
            count2 = primeFac2.count(n)
            
            for i in range(count2):
                ans *= n
                    
            done.append(n)
    
    # precondition
    assert isinstance(ans,int) and (ans >= 0), \
    "'ans' must been from type int and positive"    
    
    return ans
    
# ----------------------------------
    
def getPrime(n):
    """
        Gets the n-th prime number.
        input: positive integer 'n' >= 0
        returns the n-th prime number, beginning at index 0
    """
    
    # precondition
    assert isinstance(n,int) and (n >= 0), "'number' must been a positive int"
    
    index = 0
    ans = 2 # this variable holds the answer
    
    while index < n:
        
        index += 1
        
        ans += 1   # counts to the next number     
        
        # if ans not prime then
        # runs to the next prime number. 
        while not isPrime(ans):
            ans += 1
    
    # precondition
    assert isinstance(ans,int) and isPrime(ans), \
    "'ans' must been a prime number and from type int"    
    
    return ans
    
# ---------------------------------------------------
    
def getPrimesBetween(pNumber1, pNumber2):
    """
        input: prime numbers 'pNumber1' and 'pNumber2'
                pNumber1 < pNumber2
        returns a list of all prime numbers between 'pNumber1' (exclusiv)
                and 'pNumber2' (exclusiv) 
    """
    
    # precondition
    assert isPrime(pNumber1) and isPrime(pNumber2) and (pNumber1 < pNumber2), \
    "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
    
    number = pNumber1 + 1 # jump to the next number
    
    ans = [] # this list will be returns.
    
    # if number is not prime then
    # fetch the next prime number. 
    while not isPrime(number):
        number += 1
    
    while number < pNumber2:
        
        ans.append(number)
        
        number += 1
        
        # fetch the next prime number. 
        while not isPrime(number):
            number += 1
            
    # precondition
    assert isinstance(ans,list) and ans[0] != pNumber1 \
    and ans[len(ans)-1] != pNumber2, \
    "'ans' must been a list without the arguments"
            
    # 'ans' contains not 'pNumber1' and 'pNumber2' !
    return ans
    
# ----------------------------------------------------

def getDivisors(n):
    """
        input: positive integer 'n' >= 1
        returns all divisors of n (inclusive 1 and 'n')
    """
    
    # precondition
    assert isinstance(n,int) and (n >= 1), "'n' must been int and >= 1"

    from math import sqrt        
        
    ans = [] # will be returned.
    
    for divisor in range(1,n+1):
        
        if n % divisor == 0:
            ans.append(divisor)
       
       
    #precondition
    assert ans[0] == 1 and ans[len(ans)-1] == n, \
    "Error in function getDivisiors(...)"
    
    
    return ans


# ----------------------------------------------------


def isPerfectNumber(number):
    """
        input: positive integer 'number' > 1
        returns true if 'number' is a perfect number otherwise false.
    """
    
    # precondition
    assert isinstance(number,int) and (number > 1), \
    "'number' must been an int and >= 1"
    
    divisors = getDivisors(number)
 
    # precondition
    assert isinstance(divisors,list) and(divisors[0] == 1) and  \
    (divisors[len(divisors)-1] == number), \
    "Error in help-function getDivisiors(...)"
    
    # summed all divisors up to 'number' (exclusive), hence [:-1]
    return sum(divisors[:-1]) == number

# ------------------------------------------------------------

def simplifyFraction(numerator, denominator):
    """
        input: two integer 'numerator' and 'denominator'
        assumes: 'denominator' != 0
        returns: a tuple with simplify numerator and denominator.
    """      
    
    # precondition
    assert isinstance(numerator, int) and isinstance(denominator,int) \
    and (denominator != 0), \
    "The arguments must been from type int and 'denominator' != 0"
    
    # build the greatest common divisor of numerator and denominator.
    gcdOfFraction = gcd(abs(numerator), abs(denominator))

    # precondition
    assert isinstance(gcdOfFraction, int) and (numerator % gcdOfFraction == 0) \
    and (denominator % gcdOfFraction == 0), \
    "Error in function gcd(...,...)"
    
    return (numerator // gcdOfFraction, denominator // gcdOfFraction)
    
# -----------------------------------------------------------------
    
def factorial(n):
    """
        input: positive integer 'n'
        returns the factorial of 'n' (n!)
    """
    
    # precondition
    assert isinstance(n,int) and (n >= 0), "'n' must been a int and >= 0"
    
    ans = 1 # this will be return.
    
    for factor in range(1,n+1):
        ans *= factor
    
    return ans
    
# -------------------------------------------------------------------
    
def fib(n):
    """
        input: positive integer 'n'
        returns the n-th fibonacci term , indexing by 0
    """  
    
    # precondition
    assert isinstance(n, int) and (n >= 0), "'n' must been an int and >= 0"
    
    tmp = 0
    fib1 = 1
    ans = 1 # this will be return
    
    for i in range(n-1):
        
        tmp = ans
        ans += fib1
        fib1 = tmp
        
    return ans
